JEE Mains · Maths · STD 12 - 7.2 definite integral
Let \(f\) be a differentiable function \(R\) to \(R\) such that \(\left| {f\,(x)\, - \,f(y)} \right|\, \le \,2\,{\left| {x - y} \right|^{\frac{3}{2}}},\) for all \(x,y\,\in R .\) If \(f\,(0)=1\) then \(\int\limits_0^1 {{f^2}\,(x)\,dx} \) is equal to
- A \(0\)
- B \(\frac {1}{2}\)
- C \(2\)
- D \(1\)
Answer & Solution
Correct Answer
(D) \(1\)
Step-by-step Solution
Detailed explanation
\(|f(x)-f(y)| \leq 2|x-y|^{3 / 2}\) divide both side by \(|x-y|\) \(\left|\frac{f(x)-f(y)}{x-y}\right| \leq 2 .|x-y|^{1 / 2}\) Apply limit \(x \rightarrow y\) \(\left| {{f^\prime }(y)} \right| \le 0\) \( \Rightarrow {f^\prime }(y) = 0 \Rightarrow f(y) = c\)…
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